As in-depth as this is, I doubt that it was this. First off, “Fly” is another, antiquated word for “Run”. When he said “Fly” he meant “Don’t try and save me, get out of here.”
Secondly, by this point the Nazgul had most likely gotten their Fellbeasts, and they can already sense the Ring. They could probably track the Fellowship on the Eagles, plus Mordor isn’t that big, and there’s Barad-dur to consider. Sauron is always watching Mordor and was able to see Frodo and Sam while they were sneaking along. It would probably be pretty hard to miss nine eagles flying in with the Ring towards Mount Doom. Sauron knows that Frodo has the Ring, and Barad-dur is very close to Mt. Doom. They would most likely have to fly past the tower to get to the mountain, and I doubt Sauron would miss that. If the Nazgul attacked, the Eagles might be forced to drop the Fellowship in the middle of hordes of Orcs that I doubt they could slip through without being spotted.
We can’t really know if he wasn’t planning on taking them to the Eagles, yes, but it would be extremely impractical and most likely more dangerous than going on foot. Faster, yes, but also more conspicuous. If Sauron can spy on people through birds and beasts, there’s a high likelihood that they’d be spotted by a seemingly unassuming flock of birds. The Fellowship had already been spotted by that flock of Dunland birds that Saruman had sent to look for them, so it wouldn’t be that hard for Sauron to do the same.